No. Let $X$ and $Y$ be Banach spaces, and set $Z=X\oplus Y$, with $\||(x,y)|\|:=\|x\|+\|y\|$. Assume that $x$ is a extreme point of $x$ with $\|x\|=1$. Then $(x,0)$ becomes an extreme point of $Z$; indeed, if 
$$(x,0)=\frac12(a,y)+\frac12(b,z)$$
for $(a,y),(b,z)$ in the unit ball of $Z$, we then have $a=x=b$, since $x$ is a extreme point, but then $1=\|x\|\leq\||(a,y)|\|\leq 1$, so $y=0$, and analogously, $z=0$. 

So, $L^2(\mathbb R)\oplus L^1(\mathbb R)$, is not a dual space, but its unit ball has extreme points.